Geodesic
On unique solvability in the problem of water waves above submerged bodies
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 38, Tome 369 (2009), pp. 143-163

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Time-harmonic motion of an ideal unbounded fluid in the presence of rigid bodies located under fluid's free surface is considered. New criteria for unique solvability of the corresponding linear boundary-value problem are suggested. These criteria are based on introduction of two compact self-adjoint integral operators and investigation of their eigenvalues and eigenfunctions. For the two-dimensional problem an algorithm is developed for finding solutions to the homogeneous problem (so-called trapped modes). Examples of numerical computations illustrating the theoretical results are given. Bibl. – 18 titles. 
@article{ZNSL_2009_369_a7,
     author = {O. V. Motygin},
     title = {On unique solvability in the problem of water waves above submerged bodies},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {143--163},
     publisher = {mathdoc},
     volume = {369},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_369_a7/}
}
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O. V. Motygin. On unique solvability in the problem of water waves above submerged bodies. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 38, Tome 369 (2009), pp. 143-163. http://geodesic.mathdoc.fr/item/ZNSL_2009_369_a7/